Such a processing machine is known, for example, from FR-A-2 873 315.
This processing machine is of the type comprising:                a casing;        a tool-holder spindle which extends along an axis and which is to carry a machine tool;        a positive feed drive mechanism for the tool-holder spindle, which mechanism is suitable for driving the spindle simultaneously in rotation about its axis relative to the casing and in translation along its axis relative to the casing in accordance with an advance or return movement; and        a system for lubricating the machine tool, comprising:                    a lubricant inlet;            a lubricant duct delimited by the spindle, the lubricant duct opening out at an end of the spindle which is to carry the machine tool, in order to supply the machine tool with lubricant; and            a lubricant-admission module connecting the lubricant inlet to the said lubricant duct.                        
The drive mechanism of such a machine is referred to as “à avance mécanique” in French or as “having positive feed” in English.
Thanks to a lubricant reservoir and a pump which are located outside the casing, the lubricating system ensures the supply of lubricant to the machine tool carried by the spindle.
Nevertheless, the lubricating system has a large space requirement.